Methods for Estimating the Standard Deviation

MACHART Statement

Methods for Estimating the Standard Deviation

When control limits are computed from the input data, four methods are available for estimating the process standard deviation $\sigma$ . Three methods (referred to as the default, MVLUE, and RMSDF) are available with subgrouped data. A fourth method is used if the data are individual measurements (see "Default Method for Individual Measurements" ).

Default Method for Subgroup Samples

This method is the default for moving average charts using subgrouped data. The default estimate of $\sigma$ is

$\hat{\sigma}=\frac{s_{1}/c_{4}(n_{1})+ ... + s_{N}/c_{4}(n_{N})}N$

where N is the number of subgroups for which $n_{i}\geq2$ , s_i is the sample standard deviation of the i^th subgroup

$s_{i} = \sqrt{ \frac{1}{n_{i} - 1} \sum^{n_i}_{j=1}(x_{ij}-\bar{X}_{i})^2}$

and

$c_{4}(n_{i})=\frac{\Gamma(n_{i}/2)\sqrt{2/(n_{i}-1)} } {\Gamma((n_{i}-1)/2)}$

Here $\Gamma(\cdot)$ denotes the gamma function, and $\bar{X}_{i}$ denotes the i^th subgroup mean. A subgroup standard deviation s_i is included in the calculation only if $n_{i}\geq2$ . If the observations are normally distributed, then the expected value of s_i is $c_{4}(n_{i})\sigma$ .Thus, $\hat{\sigma}$ is the unweighted average of N unbiased estimates of $\sigma$ . This method is described in the ASTM Manual on Presentation of Data and Control Chart Analysis (1976).

MVLUE Method for Subgroup Samples

If you specify SMETHOD=MVLUE, a minimum variance linear unbiased estimate (MVLUE) is computed for $\sigma$ . Refer to Burr (1969, 1976) and Nelson (1989, 1994). The MVLUE is a weighted average of N unbiased estimates of $\sigma$ of the form s_i/c₄(n_i), and it is computed as

$\hat{\sigma}=\frac{h_{1}s_{1}/c_{4}(n_{1})+ ... + h_{N}s_{N}/c_{4}(n_{N})}{h_{1}+ ... +h_{N}}$

where

h_i = [([c₄(n_i)]²)/(1 - [c₄(n_i)]²)]

A subgroup standard deviation s_i is included in the calculation only if $n_{i}\geq2$ , and N is the number of subgroups for which $n_{i}\geq2$ .The MVLUE assigns greater weight to estimates of $\sigma$ from subgroups with larger sample sizes, and it is intended for situations where the subgroup sample sizes vary. If the subgroup sample sizes are constant, the MVLUE reduces to the default estimate.

RMSDF Method for Subgroup Samples

If you specify SMETHOD=RMSDF, a weighted root-mean-square estimate is computed for $\sigma$ as follows:

$\hat{\sigma} = \frac{\sqrt{(n_{1} - 1)s_1^2 + ... + (n_{N} - 1)s_{N}^2}} {c_{4}(n)\sqrt{n_{1} + ... + n_{N} - N}}$

The weights are the degrees of freedom n_i-1. A subgroup standard deviation s_i is included in the calculation only if $n_{i}\geq2$ ,and N is the number of subgroups for which $n_{i}\geq2$ .

If the unknown standard deviation $\sigma$ is constant across subgroups, the root-mean-square estimate is more efficient than the minimum variance linear unbiased estimate. However, in process control applications it is generally not assumed that $\sigma$ is constant, and if $\sigma$ varies across subgroups, the root-mean-square estimate tends to be more inflated than the MVLUE.

Default Method for Individual Measurements

When each subgroup sample contains a single observation ( $n_{i} \equiv 1$ ), the process standard deviation $\sigma$ is estimated as

$\hat{\sigma}=\sqrt{\frac{1}{2(N-1)} \sum_{i=1}^{N-1}{(x_{i+1}-x_{i})^2}}$

where N is the number of observations, and x₁,x₂, ... ,x_N are the individual measurements. This formula is given by Wetherill (1977), who states that the estimate of the variance is biased if the measurements are autocorrelated.

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